9 Answers
9

You should think of coverings of manifolds as analogous to field extensions. Once you accept this, then the fundamental group and absolute Galois group play the same role; coverings correspond to subgroups of the former and field extensions to subgroups of the latter (though for the absolute Galois group you have to consider its topology).

This can be made precise in algebraic geometry: if you have a covering map of projective algebraic varieties, then the function field of the target embeds into the function field of the domain by pullback, and this is a finite degree unramified field extension.

You can think of lifting paths downstairs as being a bit like algebraic number theory: each closed path downstairs has an inverse image that's a union of paths. If the covering is Galois, then each component will cover the original with the same degree, but otherwise maybe not. You can think of the conjugacy class of the path as the "Frobenius" whose orbit type on the set of preimages of a point determines the "splitting into primes."

There's even a version of the theory of L-functions given by considering the spectrum of the Laplacian for a metric on the varieties.

I saw this question a while ago and felt something in the way of a (probably misguided) missionary zeal to make at least a few elementary remarks. But upon reflection,
it became clear that even that would end up rather long, so it was difficult to find the time until now.

The point to be made is a correction: fundamental groups in arithmetic geometry are not the same as Galois groups, per se. Of course there is a long tradition of
parallels between Galois theory and the theory of covering spaces, as when Takagi writes of being misled by
Hilbert in the formulation of class field theory essentially on account of
the inspiration from Riemann surface theory. And then, Weil was fully aware that
homology and class groups are somehow the same, while speculating that a sort of
non-abelian number theory informed by the full theory of the 'Poincare
group' would become an ingredient of many serious arithmetic investigations.

A key innovation of Grothendieck, however, was the formalism for refocusing attention on the
base-point. In this framework, which I will review briefly below, when one says
$$\pi_1(Spec(F), b)\simeq Gal(\bar{F}/F),$$
the base-point in the notation is the choice of separable closure
$$b:Spec(\bar{F})\rightarrow Spec(F).$$
That is,

Galois groups are fundamental groups with generic base-points.

The meaning of this is clearer in the Galois-theoretic interpretation of the fundamental group of
a smooth variety $X$. There as well, the choice of a separable closure
$k(X)\hookrightarrow K$ of the function field $k(X)$ of $X$ can be viewed as a base-point
$$b:Spec(K)\rightarrow X$$
of
$X$, and then
$$\pi_1(X,b)\simeq Gal(k(X)^{ur}/k(X)),$$
the Galois group of the maximal sub-extension $k(X)^{ur}$ of $K$ unramified over $X$.
However, it would be quite limiting to take this last object as the definition of the fundamental group.

We might recall that even in the case of a path-connected pointed topological space $(M,b)$ with universal covering space $$M'\rightarrow M,$$
the isomorphism $$Aut(M'/M)\simeq \pi_1(M,b)$$ is not canonical. It comes rather
from the choice of a base-point lift $b'\in M'_b$. Both $\pi_1(M,b)$ and $Aut(M'/M)$
act on the fiber $M'_b$, determining bijections
$$\pi_1(M,b)\simeq M'_b\simeq Aut(M'/M)$$
via evaluation at $b'$. It is amusing to check that the isomorphism of groups obtained thereby is independent of
$b'$ if and only if the fundamental group is abelian. The situation here is an instance of the choice involved in the isomorphism
$$\pi_1(M,b_1)\simeq \pi_1(M,b_2)$$
for different base-points $b_1 $ and $b_2$.
The practical consequence is that when fundamental groups are equipped with natural
extra structures coming from geometry, say Hodge structures or Galois actions, different base-points give rise to enriched groups that are
are often genuinely non-isomorphic.

A more abstract third group is rather important in the general discussion of base-points. This is
$$Aut(F_b),$$
the automorphism group of the functor
$$F_b:Cov(M)\rightarrow Sets$$
that takes a covering $$N\rightarrow M$$ to its fiber $N_b$. So elements of $Aut(F_b)$ are
compatible collections $$(f_N)_N$$ indexed by coverings $N$ with each $f_N$ an automorphism of the set $N_b$.
Obviously, newcomers might wonder where to get such compatible collections, but
lifting loops to paths defines a natural map
$$\pi_1(M,b)\rightarrow Aut(F_b)$$
that turns out to be an isomorphism. To see this, one uses again the fiber
$M'_b$ of the universal covering space, on which both groups act compatibly.
The key point is that while $M'$ is not
actually universal in the category-theoretical sense, $(M',b')$ is universal
among pointed covers. This is enough to show that an element of $Aut(F_b)$ is completely determined by its action
on $b'\in M'_b$, leading to another bijection $$Aut(F_b)\simeq M'_b.$$
Note that the map $\pi_1(M,b)\rightarrow Aut(F_b)$ is entirely canonical,
even though we have used the fiber $M'_b$ again to prove bijectivity, whereas the identification with $Aut(M'/M)$
requires the use of $(M'_b,b')$ just for the definition.

Among these several isomorphic groups, it is $Aut(F_b)$ that ends up most relevant for the
definition of the etale fundamental group.

So for any base-point $b:Spec(K)\rightarrow X$ of a connected scheme
$X$ (where $K$ is a separably closed field, a 'point' in the etale theory), Grothendieck defines
the 'homotopy classes of etale loops' as
$$\pi^{et}_1(X,b):=Aut(F_b),$$
where $$F_b:Cov(X) \rightarrow \mbox{Finite Sets}$$ is the functor that sends a finite etale covering
$$Y\rightarrow X$$ to the fiber $Y_b$. Compared to a construction like
$Gal(k(X)^{ur}/k(X))$, there are three significant advantages to this definition.

(1) One can easily consider small base-points, such as might come from
a rational point on a variety over $\mathbb{Q}$.

(2) It becomes natural to study the variation of $\pi^{et}_1(X,b)$ with $b$.

(3) There is an obvious extension to path spaces $$\pi^{et}_1(X;b,c):=Isom(F_b,F_c),$$ making up a two-variable
variation.

This last, in particular, has no analogue at all in the Galois group approach to
fundamental groups. When $X$ is a variety over $\mathbb{Q}$, it becomes possible, for example, to study $\pi^{et}_1(X,b)$ and
$\pi^{et}_1(X;b,c)$ as sheaves on $Spec(\mathbb{Q})$, which encode rich information about
rational points. This is a long story, which would be rather tiresome to expound upon here
(cf. lecture at the INI ).
However, even a brief contemplation of it might help you to appreciate the arithmetic perspective that general $\pi^{et}_1$'s
are substantially more powerful than Galois groups. Having read thus far, it shouldn't surprise you that
I don't quite agree with
the idea explained, for example, in this post
that a Galois group is only
a 'group up to conjugacy'. To repeat yet again, the usual Galois groups are just fundamental groups with specific large base-points.
The dependence on these base-points as well as a generalization to small base-points
is of critical interest.

Even though the base-point is very prominent in Grothendieck's definition, a curious fact is that it took quite a long time for even the experts to fully metabolize its significance.
One saw people focusing mostly on base-point independent constructions
such as traces or characteristic polynomials associated to representations. My impression is that the initiative for allowing the base-points a truly active role
came from Hodge-theorists like Hain, which then was taken up by arithmeticians like Ihara and Deligne.
Nowadays, it's possible to give entire lectures just about base-points, as Deligne has actually done on several occasions.

Here is a puzzle that I gave to my students a while ago: It has been pointed out that
$Gal(\bar{F}/F)$ already refers to a base-point in the Grothendieck definition. That is,
the choice of $F\hookrightarrow \bar{F}$ gives at once a universal covering space and a base-point.
Now, when we turn to the manifold situation $M'\rightarrow M$, a careful reader may have noticed a hint above that there is
a base-point implicit in $Aut(M'/M)$ as well.
That is, we would like to write $$Aut(M'/M)\simeq \pi_1(M,B)$$ canonically for some base-point $B$. What is $B$?

Added:

-In addition to the contribution of Hodge-theorists, I should say that Grothendieck himself urges attention to many base-points in his writings from the 80's, like 'Esquisse d'un programme.'

-I also wanted to remark that I don't really disagree with the point of view in JSE's answer either.

Added again:

This question reminds me to add another very basic reason to avoid the Galois group as a definition of $\pi_1$. It's rather tricky to work out the functoriality that way, again because the base-point is de-emphasized. In the $Aut(F_b)$ approach, functoriality is essentially trivial.

Added, 27 May:

I realized I should fix one possible source of confusion. If you work it out, you find that the bijection $$\pi_1(M,b)\simeq M'_b\simeq Aut(M'/M)$$ described above is actually an anti-isomorphism. That is, the order of composition is reversed. Consequently, in the puzzle at the end, the canonical bijection $$Aut(M'/M)\simeq \pi_1(M,B)$$ is an anti-isomorphism as well. However, another simple but amusing exercise is to note that the various bijections with Galois groups, like $$\pi_1(Spec(F), b)\simeq Gal(\bar{F}/F),$$ are actually isomorphisms.

Added, 5 October:

I was asked by a student to give away the answer to the puzzle. The crux of the matter is that
any continuous map $$B:S\rightarrow M$$ from a simply connected set $S$ can be used as a base-point for the
fundamental group. One way to do this to use $B$ to get a fiber functor
$F_B$ that associates to a covering $$N\rightarrow M$$the set of splittings of the covering $$N_B:=S\times_M N\rightarrow S$$ of $S$.
If we choose
a point $b'\in S$, any splitting is determined by its value at $b'$, giving
a bijection of functors
$F_B=F_{b'}=F_b$ where $b=B(b')\in M$. Now, when $$B:M'\rightarrow M$$ is the universal
covering space, I will really leave it as a (tautological) exercise
to exhibit a canonical anti-isomorphism
$$Aut(F_B)\simeq Aut(M'/M).$$ The 'point' is that $$F_B(M')$$ has a canonical
base-point that can be used for this bijection.

Wow, thanks for the long and thoughtful answer!
–
Harold WilliamsMay 15 '10 at 0:02

2

I like the puzzle you offer very much - it leads to something far less ad-hoc than other techniques I have seen, e.g. taking the fundamental group with "basepoint" being a contractible subspace and delving into issues about loop-multiplication.
–
Tyler LawsonMay 26 '10 at 12:46

If you're interested, there is a beautiful book by Tamas Szamuely entitled Galois Groups and Fundamental Groups, which you can find here. It begins by looking at Galois groups, fundamental groups, and monodromy groups of Riemann surfaces (hence requiring only basic algebra, topology, and complex analysis) and the commonalities between them. It eventually generalizes all of this with Etale fundamental groups of schemes (this later section cites some results from commutative algebra and algebraic geometry).

I should say that I've only barely started this book, but I've heard great things about it.
–
David CorwinNov 1 '09 at 17:44

You also might be interested in the more elementary book, Abel's Theorem in Problems and Solutions, which proves the unsolvability of the quintic using monodromy groups, which are similar to fundamental groups.
–
David CorwinNov 26 '09 at 14:50

Given a (connected and semilocally 1-connected) topological space X, consider the category of not-necessarily-connected covering spaces of X. Then this category is equivalent to the category of sets equipped with an action of the fundamental group of X.

Given a field F, consider the category of finite etale F-algebras (each of which is isomorphic to a product of copies of separable extensions of F). Then this category is equivalent to the category of finite sets equipped with a continuous action of the Galois group of F.

In the latter case you can replace a field F with, say, a ring or a scheme, and ask about finite etale covers, and you get a similar equivalence involving the so-called etale fundamental group of the object. In the case of algebraic varieties over the complex numbers, these finite etale maps really are finite covering maps, and so the two have a common generalization in some sense. Over a non-algebraically closed field you actually find that the etale fundamental group is an extension of the Galois group of the base field by the "geometric" fundamental group.

If you're interested in more I'd recommend doing some reading on Grothendieck's theory of "dessin d'enfants".

(One irritating thing to note about this relationship is that the standard convention for multiplying elements in the fundamental group happens in the opposite composition order to multiplying elements in the Galois group.)

The etale fundamental group of Spec K (with K a field) is the Galois group of K. Also, for a variety X over C the profinite completion of the fundamental group of X (with the complex, not Zariski, topology) is isomorphic to the etale fundamental group of X.

You might like the following formal statement. Consider the field F with Galois group Gal. Then (finite) unramified extensions E/F are in 1-1 correspondence with (transitive) actions of Gal on (finite) sets. If you know what category is and what the Spec F is, you'll prove something like

Now perhaps you won't be familiar with all the words and symbols above. Don't worry — what is says has a direct geometric meaning. Consider indeed the (complete smooth) algebraic curve X. Let G be its fundamental group. Then in geometry you have

category of G-reps on sets = category of algebraic coverings over X

It's a very geometric exercise about the definition of fundamental group.

See, the ideas are similar and the formulas are the same — one of amazing things about mathematics is that exact theorems are often deeper and more general then handwaving.

The previous answers are very good, and I too found them useful, however I would like to point out another relation in the same vein as Dr. Tao's. Thus, this is not an answer to your question, but something that you may wish to notice if you are thinking along these lines.

This ?short? set of facts is a categorical way to think of representations as fields and the corresponding Galois groups. Notably, we get something that looks like a "Galois Correspondence" which gives an isomorphism between the group and the tensor automorphisms. This is Tannaka-Krein Theorem, which has many nice applications. I attempt here to be brief, but give you the ideas, I apologize for mistakes.

*Proposition*Consider a monoidal categories of k-representations of a compact group G, the category of k-vector-spaces and the functor, F, s.t.

(Rep_k(G), *_k,k^1)---->(Vect_k, *_k, k^1),

where k^1 is the unit object with trivial representation. There exists F which forgets the representation in the following way;

More work has been done on the (lower tech) case of L-functions and spectra of graphs (so roughly, singular one-dimensional spaces). For instance, the concept is discussed in loving detail in a sequence of papers by Harold Stark and Audrey Terras.
–
Pete L. ClarkMay 4 '10 at 22:16